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Tangential differentiated equations admit classical difference quotients

Statement

Fix $0 < α < 1$ and an integer $m \geq 4$ . Let $w \in C^{m, α}$ solve the flattened smooth quasilinear equation $A^{i j} (x, w, D w) w_{i j} + B (x, w, D w) = 0$ , with zero height on the flat boundary. Work locally, shrinking the boundary chart to a half-ball $B_{R}^{+} = {x \in R^{3} : ∣ x ∣ < R, x_{3} > 0}$ whose closed graph-jet image lies in the smooth coefficient domain, and use $C^{m, α} (\overline{B_{R}^{+}})$ for regularity up to that boundary. Repeated indices are summed. For a tangential multi-index $I$ of length $m - 2$ , set $v = D^{I} w$ . Then $v \in C^{2, α} (\overline{B_{R}^{+}})$ , has zero trace on $x_{3} = 0$ , and satisfies the classical linear equation $a^{i j} v_{i j} = f_{I}$ , where $a^{i j} = A^{i j} (x, w, D w)$ and $f_{I} \in C^{1, α} (\overline{B_{R}^{+}})$ .

Fix $0 < r < R$ and $h_{0} = (R - r) / 2$ . For a tangential coordinate unit vector $e_{t}$ and every $0 < ∣ h ∣ < h_{0}$ , the quotient $δ_{h} v (x) = (v (x + h e_{t}) - v (x)) / h$ is defined on $\overline{B_{r}^{+}}$ , belongs to $C^{2, α} (\overline{B_{r}^{+}})$ , has zero flat-boundary trace, and satisfies $a^{i j} (x) (δ_{h} v)_{i j} (x) = δ_{h} f_{I} (x) - (δ_{h} a^{i j}) (x) v_{i j} (x + h e_{t}) .$ Here $δ_{h}$ always uses the same direction and shift. The right side, $δ_{h} v$ , and each first derivative of $δ_{h} v$ have uniformly bounded $C^{0, α} (\overline{B_{r}^{+}})$ norms for $0 < ∣ h ∣ < h_{0}$ . The corresponding equation and uniform bounds hold for arbitrary multi-indices $I$ of length $m - 2$ and any fixed unit direction $e$ on nested interior balls $B_{r} \subset B_{R}$ in the coordinate domain, again for $0 < ∣ h ∣ < (R - r) / 2$ ; no boundary trace is asserted in the interior case.

Facts & Assumptions

[F1]

The flattened equation has smooth finite-dimensional coefficients on the local spacelike-jet domain, with smooth local extensions at the flat boundary, a positive principal matrix on compact admissible graph-jet neighborhoods, and homogeneous flat-boundary height; existing tangential derivatives have zero trace. Flattening a smooth fixed boundary preserves the elliptic graph equation

[F2]

The Hölder supplier defines the derivative suprema, top-order Hölder seminorm, and continuous up-to-boundary derivative convention. Product, composition, and quotient bounds used below are proved here from these definitions. Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains

[F3]

Multi-indices and $C^{k}$ regularity have their coordinate meanings; the scalar product rule applied on coordinate lines can be iterated to differentiate finite products. Cᵏ maps and multi-index derivative notation in Euclidean space Sum, product, quotient, and chain rules from difference quotients

[F4]

Continuous partial derivatives give total differentiability, and the total chain rule differentiates a smooth finite-dimensional coefficient evaluated on a $C^{1}$ graph-jet map. If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)

[F5]

Continuous second partial derivatives commute; applying this to lower derivative fields allows reordering at every derivative order already possessed by $w$ . Clairaut--Schwarz theorem for continuous second partial derivatives

[F6]

The real mean value theorem applies to a scalar function continuous on a closed segment and differentiable in its interior. We use it on coordinate segments, on segments in a convex coefficient neighborhood, and on differences of two translated scalar functions. The mean value theorem, as the case g(x) = x of Cauchy's: for f continuous on [a,b] with a < b and differentiable on (a,b) there is c ∈ (a,b) with f(b) - f(a) = f'(c)(b-a)

[F7]

Continuous real-valued functions on a nonempty compact space attain finite maxima and minima. A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism

[F8]

Closed bounded subsets of finite-dimensional Euclidean space are compact. A subset of ℝⁿ with the product topology is compact exactly when it is closed and bounded, the product topology being the Euclidean metric topology

Proof

Given: The hypotheses, chart, and nested radii in the Statement.

Goal: Prove the differentiated equation and the uniform bounds without assuming any additional derivative of $w$ .

1.1F2F3F4F5F6givenalgebra

Write $[u]_{α} = \sup_{x \neq y} ∣ u (x) - u (y) ∣ / ∣ x - y ∣^{α}$ and $∥ u ∥_{0, α} = ∥ u ∥_{\infty} + [u]_{α}$ on the domain in question. The identity $p (x) q (x) - p (y) q (y) = p (x) (q (x) - q (y)) + q (y) (p (x) - p (y))$ gives $[p q]_{α} \leq ∥ p ∥_{\infty} [q]_{α} + ∥ q ∥_{\infty} [p]_{α}$ and hence $∥ p q ∥_{0, α} \leq ∥ p ∥_{0, α} ∥ q ∥_{0, α}$ . Iterating [F3] and using [F5] expresses every derivative of a product as a finite sum of products of derivative fields. On a bounded convex ball or half-ball, a derivative field with bounded first derivatives is Lipschitz by the chain rule along segments and [F6], and therefore is Hölder with seminorm at most its Lipschitz constant times the diameter to the power $1 - α$ . The same estimates extend to the closure by continuity from interior points. Thus all lower derivatives as well as the top derivatives of a $C^{k, α}$ field are Hölder, and the displayed product inequality applied to the differentiated products proves closure under finite products in $C^{k, α}$ .

1.2F2F3F6givenalgebra

For $x \in \overline{B_{r}^{+}}$ and $0 < ∣ h ∣ < h_{0}$ , every $x + s h e_{t}$ , $0 \leq s \leq 1$ , has norm at most $r + ∣ h ∣ < R$ and the same normal coordinate as $x$ . Thus both endpoints and the connecting segment lie in the original chart, including its flat boundary when $x_{3} = 0$ . For any scalar $u \in C^{1, α} (\overline{B_{R}^{+}})$ and any interior point $x$ , apply [F6] to $t \mapsto u (x + t e_{t})$ on the interval with endpoints $0, h$ to obtain $∣ δ_{h} u (x) ∣ \leq ∥ \partial_{t} u ∥_{\infty}$ . For two interior points $x, y \in B_{r}^{+}$ apply it instead to $t \mapsto u (x + t e_{t}) - u (y + t e_{t})$ on the same interval. Its derivative is $\partial_{t} u (x + t e_{t}) - \partial_{t} u (y + t e_{t})$ , whose absolute value is at most $[\partial_{t} u]_{α; B_{R}^{+}} ∣ x - y ∣^{α}$ . Consequently $[δ_{h} u]_{α; B_{r}^{+}} \leq [\partial_{t} u]_{α; B_{R}^{+}}$ and $∥ δ_{h} u ∥_{0, α; B_{r}^{+}} \leq ∥ \partial_{t} u ∥_{0, α; B_{R}^{+}}$ . These bounds pass to flat-boundary points by continuous extension. This proves the uniform quotient estimate for both signs of $h$ directly, with no differentiability beyond $C^{1, α}$ .

2.1F1F2F3F4F5F6F7F8step 1.1choose

Put $G (x) = (x, w (x), D w (x))$ , which is $C^{m - 1, α}$ by [F2] and [F5]. Use the smooth local extensions from [F1] at a boundary jet, and choose a closed coefficient-space ball $K$ centered at that jet, contained in an open ball on which those extensions are smooth. By continuity of $G$ , shrink the chart so that $G (\overline{B_{R}^{+}}) \subset K$ . The ball $K$ is convex and compact by [F8]. For any smooth scalar coefficient $F$ among these extensions, every derivative of $F$ needed here and its next derivative are bounded on $K$ by [F7]. Applying [F4] and [F6] on coefficient-space segments makes each such derivative of $F$ Lipschitz on that ball. Thus its composite with $G$ is Hölder, since $G$ is Hölder. Repeatedly applying [F3] and [F4] expresses a derivative of order $j \leq m - 1$ of $F \circ G$ as a finite sum of a derivative of $F$ evaluated at $G$ times derivative fields of $G$ of orders at most $j$ . The estimates in step 1.1 show that all these terms are Hölder and bounded; their continuous boundary extensions are given by the same expressions. In particular $a^{i j}$ and $b (x) = B (x, w, D w)$ are $C^{m - 1, α}$ locally up to the flat boundary.

3.1F1F2F3F5step 1.1step 2.1algebra

Differentiate $a^{i j} w_{i j} + b = 0$ by $D^{I}$ , using the iterated product rule and commuting only derivatives already supplied by $w \in C^{m, α}$ . With $J \leq I$ denoting componentwise order and $(\binom{I}{J})$ the product of the ordinary binomial coefficients, the result is $a^{i j} v_{i j} = f_{I}$ , where $f_{I} = - D^{I} b - \sum_{0 < J \leq I} (\binom{I}{J}) (D^{J} a^{i j}) D^{I - J} w_{i j}$ . In every summand $1 \leq ∣ J ∣ \leq m - 2$ , so $D^{J} a^{i j}$ is at least $C^{1, α}$ by step 2.1, while $D^{I - J} w_{i j}$ has order $m - ∣ J ∣ \leq m - 1$ and is at least $C^{1, α}$ . Also $D^{I} b \in C^{1, α}$ . Step 1.1 therefore gives $f_{I} \in C^{1, α}$ . All $m$ derivatives needed for the left side already exist, so the equation is classical; continuity extends it to the flat boundary. Finally $v \in C^{2, α}$ follows from the definition and [F5], and its zero trace follows from [F1] because $I$ is tangential.

4.1F2F3step 1.1step 2.1step 3.1step 1.2algebra

For each permitted fixed nonzero $h$ , translation and subtraction preserve $C^{2, α}$ regularity, and $D^{β} δ_{h} v = δ_{h} D^{β} v$ for $∣ β ∣ \leq 2$ by differentiation of the two translates. Tangential shifts preserve the zero trace. Subtracting $a^{i j} (x) v_{i j} (x) = f_{I} (x)$ from the equation at $x + h e_{t}$ , and dividing by $h$ , gives exactly the displayed equation after writing the shifted product difference as $a^{i j} (x) δ_{h} v_{i j} + δ_{h} a^{i j} (x) v_{i j} (x + h e_{t})$ . Step 1.2 applied to $f_{I}$ , to each $a^{i j}$ , to $v$ , and to each $v_{j}$ bounds $δ_{h} f_{I}$ , $δ_{h} a^{i j}$ , $δ_{h} v$ , and $D δ_{h} v$ uniformly in $C^{0, α}$ . Translation does not enlarge the Hölder norm of $v_{i j}$ restricted from the larger chart, so step 1.1 bounds each product on the right uniformly. This uses $v_{j} \in C^{1, α}$ and $v_{i j} \in C^{0, α}$ , never a third derivative of $v$ .

5.1F2F3F4F6step 1.1step 2.1step 3.1step 1.2step 4.1algebra∎

On nested interior balls the norm bound $∣ x + s h e ∣ \leq r + ∣ h ∣ < R$ holds for any fixed unit direction $e$ , with no restriction on $I$ . The composition, product, and differentiated-equation arguments are unchanged. For the quotient estimate replace $\partial_{t} u$ by $D_{e} u$ , a fixed finite linear combination of first coordinate derivatives; it has the required $C^{0, α}$ bound. Thus the same mean value calculation and product bound prove the interior assertion. Together these steps establish every claimed equation, trace, and uniform bound on the stated small-shift domains.

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